# Full text of "Mathematical And Physical Papers - Iii"

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```ON  THE  CONDUCTION OF HEAT IN  CRYSTALS.            221
steady, and thus we fall back on the case first considered. We may conclude therefore generally, that if the bar be sufficiently slender, the direction of the maximum flux, even close to the surface, will sensibly coincide with that of the length of the bar; so that the isothermal surfaces, which are necessarily perpendicular to the direction of the flow of heat, will be planes perpendicular to the axis of the bar.
By supposing the bar to be the auxiliary solid belonging to a crystalline bar, we arrive at the following theorem. If a slender crystalline bar be heated at one end, and if we confine our attention to points of the bar situated at a sufficient distance from the source of heat to render insensible any irregularities attending the mode of communicating the heat, the isothermal surfaces will be sensibly planes parallel to the diametral plane of the thermic ellipsoid which is conjugate to the system of chords drawn parallel to the length of the bar. These planes will necessarily have an oblique position unless the direction of the length of the bar be a thermic axis of the crystal.
The same result might have been obtained without employing the auxiliary solid, by first shewing that when the bar is sufficiently slender the direction of the flow of heat sensibly coincides with that of the length of the bar. We should thus be led to a problem exactly the converse of that treated in Art. 13, namely, Given the direction of the flow of heat, to find that of the isothermal surface.
16. It may be shewn in a similar way, that if a thin plate be formed of an uncrystallized substance, and be heated at one or more places, or over a finite portion, if we consider only those parts of the plate which are situated at a sufficient distance from the sources of heat to render insensible any irregularities attending the mode in which the heat is communicated, the flow of heat will take place in a direction sensibly parallel to the plate, and therefore the isothermal surfaces will be cylindrical surfaces whose generating lines are perpendicular to the plate. It is here supposed that the lateral boundaries of the plate are situated at a sufficient distance to render their effect insensible.
Hence, in a thin crystalline plate heated in a similar manner, the isothermal surfaces, under similar restrictions, will be cylin-```